Some bivariate stochastic models arising from group representation theory

Iglesia, Manuel D. de la; Román, Pablo

doi:10.1016/j.spa.2017.10.017

Cited by 7 publications

(4 citation statements)

References 35 publications

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“…Certain families of MVOP related to the pair (SU(n + 1), U(n)) have been exploited to derive stochastic models, see [8,10,13]. The family described in Section 3 leads to models of continuous-time bivariate Markov processes which are analyzed in detail in [14].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs

van Pruijssen,

Román

2016

Preprint

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In this paper we present a method to obtain deformations of families of matrixvalued orthogonal polynomials that are associated to the representation theory of compact Gelfand pairs. These polynomials have the Sturm-Liouville property in the sense that they are simultaneous eigenfunctions of a symmetric second order differential operator and we deform this operator accordingly so that the deformed families also have the Sturm-Liouville property. Our strategy is to deform the system of spherical functions that is related to the matrix-valued orthogonal polynomials and then check that the polynomial structure is respected by the deformation. Crucial in these considerations is the full spherical function Ψ 0 , which relates the spherical functions to the polynomials. We prove an explicit formula for Ψ 0 in terms of Krawtchouk polynomials for the Gelfand pair (SU(2) × SU(2), diag(SU(2))). For the matrix-valued orthogonal polynomials associated to this pair, a deformation was already available by different methods and we show that our method gives same results using explicit knowledge of Ψ 0 .Furthermore we apply our method to some of the examples of size 2 × 2 for more general Gelfand pairs. We prove that the families related to the groups SU(n) are deformations of one another. On the other hand, the families associated to the symplectic groups Sp(n) give rise to a new family with an extra free parameter.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs

van Pruijssen,

Román

2016

Preprint

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“…Therefore D ∈ B 2 R (P). For the converse let D ∈ B 2 R (P) so that 13), we obtain the following recurrence relation for Qx :…”

Section: Moreover the Dual Sequence Satisfies The Three Term Recurren...mentioning

confidence: 99%

“…In [41] it is shown that the only orthogonal polynomials that have orthogonal duals are the Askey-Wilson polynomials and limiting or sub-families. The theory of matrix valued orthogonal polynomials (MVOP) was initiated by Krein [40] and has connections and applications in different areas of mathematics and mathematical physics such as scattering theory [25], tiling problems [15], integrable systems [3,5,6,31], spectral theory [26] and stochastic processes [12,13,28].…”

Section: Introductionmentioning

confidence: 99%

Duality and Difference Operators for Matrix Valued Discrete Polynomials on the Nonnegative Integers

2023

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In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators acting on the matrix orthogonal polynomials. These operators belong to the so called Fourier algebras, which play a key role in the construction of the families. In order to illustrate duality, we describe a family of Charlier type matrix orthogonal polynomials with explicit shift operators which allow us to find explicit formulas for three term recurrences, difference operators and squared norms. These are the essential ingredients for the construction of different dual families.

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“…Matrix valued orthogonal polynomials (MVOPs) were introduced by Krein in the 1940s and they appear in different areas of mathematics and mathematical physics, including spectral theory, 1 scattering theory, 2 tiling problems, 3 integrable systems, 4–7 and stochastic processes 8–10 . There is also a fruitful interaction between harmonic analysis of matrix valued functions on compact symmetric pairs and MVOPs.…”

Section: Introductionmentioning

confidence: 99%

Ladder relations for a class of matrix valued orthogonal polynomials

2020

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Using the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on double-struckR, and we derive algebraic and differential relations for these MVOPs. A particular case of importance is that of MVOPs with respect to a matrix weight of the form Wfalse(xfalse)=e−v(x)exAexA* on the real line, where v is a scalar polynomial of even degree with positive leading coefficient and A is a constant matrix.

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Some bivariate stochastic models arising from group representation theory

Cited by 7 publications

References 35 publications

Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs

Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs

Duality and Difference Operators for Matrix Valued Discrete Polynomials on the Nonnegative Integers

Ladder relations for a class of matrix valued orthogonal polynomials

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